Maths
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To kick-start this topic:
MAGIC SQUARES
Definition: A magic square is defined to be a square of numbers such that each row, column and (major) diagonal has the same sum. As an example,
2 9 4
7 5 3
6 1 8
has a sum of 15 along each row/column/diagonal.Problems: Complete the following magic squares.
#1
02 12 10
?? 08 ??
?? ?? ??#2
13 20 ??
?? 14 ??
?? ?? ??#3
Show that if a, b and c are known below then there is a unique solution:
a b ?
? c ?
? ? ?MAGIC SQUARES – PART 2
(P.S. Feel free to extend/reply as you wish!)
#4
Show that the following square has no solutions.
1 2 3
? 4 ?
? ? ?#5
Find x below.
12 ?? ??
?? x ??
?? ?? 02#6
Complete this magic square.
15 01 ??
?? ?? 13
?? ?? ??1:02,12,10
16,08,00
06,04,142:13,20,09
10,14,18
19,08,154: [possible numbers are 1,2,3,4,5,6,7,8,9]
top row: 1+2+3 = 6
using this, this would mean that 1+4+bottom right square should be 6
however, the only possibility the the bottom right square is 1, which is already used
therefore, the square has no solutions5: x=7
6:15,01,11
05,09,13
07,17,03The problem of constructing magic squares with prime numbers only was first discussed by Henry E. Dudeney in The Weekly Dispatch for 22nd July and 5th August 1900. Dudeney went on to describe a 3rd order magic square comprised of nine different prime numbers* with the lowest possible constants.
Can you find it?
*Dudeney included the number 1 as a prime number for this exercise.
@AB1, nice solutions. For #4, it turns out that even if repeat numbers were valid, there still wouldn’t be a solution. Can you find out why?
@Rhydwen23, I have that any magic square of primes can only consist of numbers that are either all 1 more than a multiple of 6 or all 1 less than a multiple of 6. Because of this I claim that the smallest solution that satisfies your conditions is:
31 73 7[Sequel to post #89569]
Sorry! Accidentally pressed the ‘Submit’ button…(!!!) The solution I had is:
31 73 07
13 37 61
67 01 43
(or equivalent)MAGIC SQUARES – PART 3
#7
Show that no 2×2 magic squares exist (other than the trivial case where all numbers in the square are equal).#8
Solve the following 4×4 magic square (repeat numbers valid) – decimals may appear!
?? ?? 20 02
?? ?? ?? 01
22 ?? ?? ??
05 10 20 23This is a bit more relevant to the National Cipher Challenge: 2002 marked the year of the (01)st Cipher Challenge. The 22nd edition will commence on 05/10/2023.
Well done @kford_academy. That is just as Dudeney had it.
The solution, for this 3rd order, and up to 12th order, magic square of prime numbers is provided in an article in the Monist (Chicago) for October 1913 by W S Andrews and H A Sayles. This is available on-line.
For anyone not wishing to include the number 1 as a prime, then I think the 3rd order magic square of prime numbers, with the lowest factor (177) is:
101 005 071
029 059 089
047 113 017INFORMATIONAL POST – MAGIC SQUARES
===================================
THE WONDER OF THE NUMBER 1480028USING THE NUMBERS: 129 141 153 159 171 183 189 201 213
WE FORM A 3 DIGIT MAGIC SQUARE WITH CONSTANT 513141 189 183
213 171 129
159 153 201
We find that if we put these numbers onto the end of 1480028 they make nine primes:
1480028129 1480028141 1480028153 1480028159 1480028171 1480028183 1480028189 1480028201 1480028213AND SO WE CAN FORM A 10 DIGIT ALL PRIME MAGIC SQUARE WITH CONSTANT 4440084513
1480028141 1480028189 1480028183
1480028213 1480028171 1480028129
1480028159 1480028153 1480028201
TLW 2 Quickies
==============Q1: I doubt you will be positive with your answer!
===================================================
What number gives the same result
when it is added to 1/2
as when it is multiplied by 1/2?Q2: The devil does math
========================
He takes HIS number and squares it
He spreads out the answer into digits
He cubes each digit and then adds them all together (answer 1)He takes HIS number and cubes it
He spreads out the answer into digits
He adds each of the digits together (answer 2)
He adds together the two answers (answer 1 + answer 2)He smiles to himself, why?
Bonus: Can you write it out as a math statement?The Devil may have all the best tunes, but he doesn’t have a monopoly on numbers of the type described by The_Letter_Wriggler in
#91127 Q2: The devil does math. Can you find another number that matches the criterion laid out in Q2, other than the Devil’s number itself?[I haven’t had time to check your calculations, but then I don’t need to.! With such a talented crew taking part, someone should be able to come up with an answer! Harry]
@Rhydwen23
Never though to test for that! But yes a four digit number N
(so as not to give it away) whose 12 divisors sum to 4368Is this right?
The Prime Counting Function (PCF) says the number of primes less than or equal to N is given by N/ln(N), where ln(N) is the natural logarithm of N. For example the Prime Counting Function suggests 21.715 primes less than or equal to 100; suggesting a probability of any single integer between 1 and 100 being prime is 21.715 / 100 = 0.21715.
For TLW’s ten-digit magic square the smallest and largest numbers are:
N = 1480028213, PCF = 70092602.11
N = 1480028129, PCF = 70092598.32A range of 84.
It seems risky to subtract two large numbers and put any faith in the small remainder, but ignoring that, the PCF suggest just 3.79 prime numbers lie between 1480028129 and 1480028213. Giving a probability of any particular single number being prime as 3.79 / 84 = 0.04512. The probability of all, of any nine numbers from within that range, being prime is (3.79/84)^9 = 7.75 x 10^-13. That seems a vanishingly small probability and suggests a remarkable result to find the case that beats the odds.
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